Electric dipole polarizabilities and C6 dipole-dipole dispersion coefficients for alkali metal clusters and C60

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Linköping University Post Print

Electric dipole polarizabilities and C6

dipole-dipole dispersion coefficients for alkali metal

clusters and C60

Auayporn Jiemchooroj, Bo. E. Sernelius and Patrick Norman

N.B.: When citing this work, cite the original article.

Original Publication:

Auayporn Jiemchooroj, Bo. E. Sernelius and Patrick Norman, Electric dipole polarizabilities and C6 dipole-dipole dispersion coefficients for alkali metal clusters and C60, 2007, Journal of Computational Methods in Sciences and Engineering, (7), 5-6, 475-488.

Copyright: IOS Press

http://iospress.metapress.com/

Postprint available at: Linköping University Electronic Press

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Electric dipole polarizabilities and

C

6

dipole-dipole dispersion

coefficients for alkali metal clusters and C

₆₀

Auayporn Jiemchooroj, Bo E. Sernelius, and Patrick Norman∗

Department of Physics, Chemistry and Biology, Link¨oping University SE-581 83 Link¨oping, Sweden

(Dated: March 6, 2007)

Abstract

The frequency dependent polarizabilities of closed-shell alkali metal clusters containing up to ten lithium, potassium, and rubidium atoms have been calculated using the linear complex polar-ization propagator approach in conjunction with Hartree–Fock and Kohn–Sham density functional theory. In combination with polarizabilities for C60 from a previous work [J. Chem. Phys. 123,

124312 (2005)], the C6 dipole-dipole dispersion coefficients for the metal cluster-to-cluster and

cluster-to-buckminster fullerene interactions are obtained via the Casimir–Polder relation. The B3PW91 results for the polarizabilities and dispersion interactions of the alkali metal dimers and tetramers are benchmarked against couple cluster calculations, and the whole series of calculations are compared against the corresponding work on sodium clusters [J. Chem. Phys. 125, 124306 (2006)]. The error bars of the reported theoretical results for the C6 coefficients are estimated to

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I. INTRODUCTION

The electric-dipole polarizability is the key microscopic property in several spectroscopies. Much theoretical and experimental work has been devoted to determine the polarization properties of atoms, molecules, and clusters,1 _{and, in recent years, this research has}

in-cluded the study of metal clusters.2 _{In particular sodium has played the role of model}

system for metal cluster research. The original measurements by Knight et al.3 _{of the}

polarizability of sodium clusters containing up to 40 atoms have spurred a large number of theoretical calculations4–11 and experiments12–14 devoted to the ground state electronic structure and optical properties of these systems. The polarization properties of lithium clus-ters have also attracted quite some attention in the literature both in the theoretical4,15–17

and experimental12,18 communities, whereas, for potassium and rubidium, earlier reports on cluster polarizabilities are restricted to the potassium dimer3,4,19_{and the rubidium dimer.}4,19

The leading long-range dispersion interaction is the van der Waals dipole-dipole interac-tion which relates to the electric dipole polarizability via the Casimir–Polder relainterac-tion.20_{It is}

a common procedure to express the strength of this interaction in terms of the C6 dispersion

coefficient. In an experiment by Kresin et al.21 _{sodium clusters were injected into a}

cav-ity containing a low concentration of C60 fullerenes and, from the measured scattering, the

cluster-fullerene long-range interaction potential could be determined. In addressing this experiment, we recently reported first-principles calculations of the dispersion interaction between C60 and sodium clusters.22 To the best of our knowledge and prior to the present

work, the potentials for long-range interactions have not been determined for alkali metal clusters containing lithium, potassium, or rubidium.

The evaluation of the C6 coefficient for a pair of microscopic systems involves the

de-termination of the dynamic polarizability at imaginary frequencies, i.e., α(iω), for each of the two individual systems. Different computational approaches have been designed for this purpose; it is, for instance, possible to determine α(iω) by turning to an expansion of the polarizability in the Cauchy moments as described in several papers.23–26 _{More recently it}

was argued that a more efficient strategy is to adopt a straightforward approach to the evaluation of α(iω) at the cost of introducing complex algebra into the time-dependent elec-tronic structure code.27–29_{A number of applications have demonstrated that accurate results}

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widened to include large-scale systems22,30 _{and excited states.}31 _{In fact the computational}

scaling parallels that of traditional polarization propagator approaches developed for the calculation of the regular polarizability depending on a real frequency argument.

In the present work we will present a systematic study of the dispersion interactions for small alkali metal (lithium, sodium, potassium, and rubidium) clusters including up to ten atoms as well as the interactions between these clusters and the C60 fullerene. We

will base our work on the aforementioned complex polarization propagator approach.27,29

The properties of the metal clusters are obtained at the Hartree–Fock (HF) and density functional theory (DFT) levels and it will be argued that the quality of the DFT results as obtained with the hybrid B3PW91 exchange-correlation functional will correspond to the second-order Møller–Plesset (MP2) wave function model. Our work is complementary to the previous investigation on sodium clusters,22 _{and, for the convenience of the reader and}

sake of completeness, parts of the results presented in Ref. 22 will be repeated here.

II. METHODOLOGY

Our approach is focused at a direct evaluation of the electric dipole polarizability α(iω) by turning to complex algebra in the time-dependent electronic structure method of choice.27,29

The polarizability corresponds to the first-order response in the molecular dipole moment as due to an external electric field; in the case of dispersion interactions between systems A and B, the external field exposed on system A is that due to the induced dipole moment in system B (and vice versa). If expressed in the basis of eigenstates to the molecular Hamiltonian, the polarizability of the molecular system evaluated for a frequency on the imaginary axis can be written in terms of a sum-over-states formula according to

ααβ(iω) = ¯h−1 X n ′(h0|ˆµ_α|nihn|ˆµ_β|0i ωn− iω + h0|ˆµβ|nihn|ˆµα|0i ωn+ iω ) , (1)

where ˆµα is the electric dipole operator along the molecular axis α, ωn is the transition

frequency of the excited state |ni, and the prime indicates omission of the ground state in the summation. Once the polarizability has been determined, the C6 dispersion coefficient

of the orientationally averaged long-range dipole–dipole interaction between systems A and B is given by20 C6 = 3¯h π Z ∞ αA(iω)αB(iω)dω, (2)

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where α denotes the trace of the polarizability tensor. It is clear that in order to carry out the integration in Eq. (2) we must, at least in principle, determine the polarizabilities of systems A and B on the entire imaginary axis. In practice, however, we can calculate α(iω) for a set of discrete frequencies and use a quadrature scheme for the integration, see Section III for details on how the integration in Eq. (2) is carried out in the present work.

Apart from the case of calculations based on the configuration interaction method in quantum chemistry, the eigenstates of the quantized Hamiltonian are not included in the excitation manifold and the sum-over-states property expressions that we meet in an exact formulation are instead represented by matrix equations. In the single determinant Hartree– Fock or Kohn–Sham approximations, the polarizability in Eq. (1) corresponds to the well known random phase approximation equation27,29

hhA; Biiiω = −A[1]†

n

E[2]− iωS[2]o−1B[1], (3) where E[2] _{and S}[2] _{are the so-called Hessian and overlap matrices, and A}[1] _{and B}[1] _{are the}

property gradients composed from the ground-to-excited state transition moments of dipole moment operators ˆµα and ˆµβ, respectively.

Equation (3) is complex, and corresponds to a set of two coupled real matrix equations. In Ref. 28 we gave explicit account of how these coupled equations could be efficiently solved, and the resulting polarizability is a well-behaved monotonic function that has its maximum in the static limit α(0) and then tends to zero as ω → ∞. The rate at which α(iω) tends to zero is sometimes expressed in terms of a so-called effective frequency ω1 that is defined in

the London approximation

α(iω) = α(0) 1 + (ω/ω1)2

, (4)

Having introduced this approximation, the evaluation of the integral for the dispersion coefficient [Eq. (2)] can be made analytically and, for like molecules, one obtains the relation

C6 =

3¯hω1

4 [ α(0) ]

2

. (5)

It was shown in Ref. 32 that the effective frequency can be regarded as more or less constant for an entire class of systems, and, for that reason, property predictions based on extrap-olation of results can sometimes be accurately made. Therefore, resulting values for the effective frequencies of the alkali metal clusters have been included in the present work.

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III. COMPUTATIONAL DETAILS

The polarizabilities at imaginary frequencies were determined from Eq. (3) at the time-dependent HF and DFT levels of theory with use of the dalton program.33_{The DFT}

calcu-lations were performed with use of the hybrid B3PW91 exchange-correlation functional.34,35

Unless specified, the polarization basis sets of Sadlej36 _{were employed in the property}

calcu-lations for all alkali metal clusters; however, for the dimer and tetramer compounds, larger basis sets37 _{[19s15p12d6f ] were also used (but then marked with a footnote in the tables).}

For lithium and sodium, all calculations refer to all-electron parametrization of the density, but, for potassium and rubidium, we used the Stuttgart effective core potentials (ECPs).38

The property calculations made with ECPs employ the all-electron polarization basis sets of Sadlej36,37 for the description of the valence orbitals, i.e., the core atomic orbitals are left to be projected out by the projectors in the ECPs.

In order to evaluate the Casimir–Polder integral [Eq. (2)] for the C6 dispersion

coeffi-cients, the polarizabilities were calculated at the imaginary frequencies taken from a Gauss– Legendre integration scheme with the transformation of variables according to

iω = iω0(1 − t)/(1 + t). (6)

Here we used a transformation factor of ω0 = 0.3Eh as suggested in Ref. 39, followed by

a Gauss–Legendre quadrature in the interval −1 ≤ t ≤ 1. For the interactions between the alkali clusters and C60, results for the polarization of the fullerene were taken from our

previous work.30 _{We use the six-point Gauss–Legendre integration in the present work.}

The coupled cluster model with single and double excitations (CCSD)26,40 _{was used to}

obtain the Cauchy moments and the C6coefficients of the alkali metal dimers and tetramers.

The frequency-dependent polarizabilities can be obtained from the Cauchy moments by the Cauchy moments expansion, and the C6dispersion coefficients can be directly evaluated from

the Cauchy moments with use of the lower [n, n−1]α and upper [n, n−1]β Pad´e approximants

recommended by Langhoff and Karplus.41 _{With n = 4, the value of the C}

6 coefficients for

the dimers and tetramers are converged to within 1%.

For the dimers Li2, Na2, K2, and Rb2, we used the experimental bond lengths of 2.6725,42

3.0788,42_3.923,42 _{and 4.18 ˚}_A,43 _{respectively; for the lithium clusters, we used the}

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enforc-ing point group symmetry); for the sodium clusters, we used the B3LYP/6-311+G(d) opti-mized structure from Ref. 10 for the tetramer and the B3LYP/6-31G(d) optiopti-mized structures from Ref. 9 for others. For the potassium and rubidium clusters, structure optimizations were carried out in the present work by use of the gaussian03 program;44 _{the geometries}

were optimized at the DFT/B3LYP level34 _{using the Stuttgart ECPs and valence basis sets}

as included in the basis set library.

IV. RESULTS AND DISCUSSION

We begin our discussion by estimating the accuracy of the theoretical results for the po-larizabilities and the C6 dipole-dipole dispersion coefficients. We then turn to a presentation

of the long-range interactions between closed-shell alkali metal clusters containing up to ten lithium, sodium, potassium, and rubidium atoms and the fullerene C60.

A. Estimating the quality of results

Apart from the diatomic systems, experimental results for the cluster structures are unavailable, and, in addition, for the potassium and rubidium clusters there are neither experimental nor theoretical geometries available. For that reason we conducted geometry optimizations for the Knand Rbn(n = 4, 6, 8, and 10) systems. For the larger clusters there

are clearly a large number of possible configurations, and we have made no attempt to find the global minima of the potassium and rubidium clusters. Instead we argue that it is plau-sible that the optimal configuration of the potassium and rubidium clusters should possess the symmetry elements of the corresponding sodium clusters for which the structure minima are reported in Refs. 9 and 10, and we therefore use the optimized sodium structures9,10 _as

initial configurations in the optimizations of potassium and rubidium clusters. The B3LYP optimized bond lengths of the Nan, Kn and Rbn clusters are given in Table I (the bond

labels are illustrated in Fig. I). The corresponding bond parameters for the Lin clusters are

not included in the table because point group symmetries were not enforced in the orig-inal molecular structure optimizations which lead to slightly asymmetric clusters.15 _This

aspect of asymmetry is also reflected by the tensor components of the polarizability that are reported in the present work (compare for instance the three α-components of Li8 in

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Table II).

The quality of our results for the fullerene and the sodium clusters was established in our previous work by a comparison to experimental results as well as, for the smaller clusters, theoretical results obtained with more sophisticated ab initio approaches, and it was con-cluded that the DFT results for α(0) and the C6 coefficient had error bars of 2% and 5%,

respectively.22,30_{In the present work, we will address the quality of the property calculations}

for the other alkali metal clusters. It is clear from the DFT calculations by Chandrakumar et al.10that the accuracy of polarizabilities for sodium clusters is significantly improved by the use of hybrid exchange-correlation functionals, and we therefore adopt the hybrid B3PW91 functional34,35 _{for the calculations of polarizabilities of alkali metal clusters.}

Experimental results have not been found for the dispersion coefficients involving lithium, potassium, or rubidium clusters, but, for the static polarizability α(0) there exists experi-mental data for the set of lithium clusters12,18 _{as well as the potassium}3,19 _{and rubidium}19

dimers. In establishing the quality of the DFT results for the polarization and dispersion properties of Lin, Kn, and Rbn, we also compare our results to theoretical results that are

obtained with wave function correlated methods. With respect to results found in the liter-ature attention is paid to the employed cluster geometries and basis sets. We address these issues by carrying out a set of coupled cluster calculations on the dimers and tetramers, so that we are in control of the computational parameters. Our wave function correlated cal-culations are of course carried out with use of identical parameter sets (geometry and basis set) as those used in our DFT calculations. We can also determine the dispersion coefficient at the coupled cluster level so that we can get an estimate of the quality of the dispersion of the polarizability α(iω) (and not only the static value) at the DFT level of theory.

In Table II, we present the results for the polarizability of the lithium dimer and tetramer and comparison is made against the results from the literature that are obtained with highly correlated wave function methods.4,15–17_{For Li}

2, the MP2 result for α(0) from Ref. 15 is 202.8

a.u. at the optimized bond length of 2.709 ˚A, which is to be compared with our CCSD result of 205.0 a.u. at the experimental bond length of 2.6725 ˚A. Our coupled cluster calculation employ a large uncontracted basis set of size [19s15p12d6f ] that should be flexible enough to provide a quite accurate CCSD result for the polarizability. The bond distances in the two compared calculations differ, however, and we have therefore also determined α(0) at a bond length of 2.709 ˚A using CCSD and the same large uncontracted basis set. The result

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of this calculation is 204.4 a.u., so we conclude that the MP2 and CCSD results are in close agreement. The CCSD and CCSD(T) results of Urban and Sadlej4 _{are 215.7 and 216.0 a.u.,}

respectively, which is significantly larger than our best CCSD result. This discrepancy is due the use of a smaller basis, as is seen in Table II where we have reproduced the CCSD result with use of Sadlej’s polarization basis set.36 _{More specifically, it is mainly the reduced}

basis set flexibility due the contractions that are the main cause for the discrepancy; with a decontracted Sadlej basis set of size [10s6p4d], Pecul et al.16 _{reported the value of α(0) to}

be 208.7 a.u. (at the same bond distance), which is in much closer agreement with our best CCSD result. The MP2 result from Ref. 15 for the tetramer is 343.7 a.u. which is in close agreement with our CCSD result of 342.4 a.u. For the tetramer we are unable to employ the larger basis set in the CCSD, but, based on the above discussion of the results for the dimer, it is reasonable to expect that this value is somewhat too large compared to the basis set limiting value. We conclude, however, that the quality of referenced MP2 results15

for polarizability of the lithium dimer and tetramer is at the level of the corresponding CCSD results with an appropriate basis set, and we consider the MP2 results for the larger lithium clusters as the best available reference for our calculations of α(0). Indeed there is a CCSD value of 540.9 a.u. reported for the polarizability of Li8,16 but the predominant

reason for the large discrepancy between this result and the MP2 value15 _{of 609.2 a.u. is}

the difference in the molecular configurations rather than the difference in the treatment of electron correlation. Since we have adopted the lithium cluster structures of Ref. 15 we choose to use the MP2 results as our reference values.

The reason for us to establish the quality of the referenced MP2 results for the polariz-ability of the lithium clusters is that these results are in close agreement with our results obtained at the Kohn–Sham DFT level of theory using the B3PW91 functional. Compar-ing our DFT results with the MP2 reference values, we see that the discrepancies in α(0) amount to 4% for the lithium tetramer and to less than 2% for others. This finding is in line with the conclusion made in Ref. 22 about the DFT/B3PW91 results for the polariz-ability of the sodium clusters namely that the quality of the DFT results parallels that of the corresponding MP2 results. In Table III we present parts of the sodium cluster results from our previous work22 _{with the addition made here of providing the results for the}

in-dividual tensor components of the polarizability. But we refer to the original work for the discussion that lead us to the conclusion that the error bar of our DFT/B3PW91 results for

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the polarizability of the sodium clusters is 2%.

When it comes to the polarizability of the potassium and rubidium clusters results in the literature are scarce, but for the two dimers there is the work of Urban and Sadlej.4

For the potassium dimer our CCSD result for the polarizability is in close agreement with their CCSD(T) result (see Table IV), but for the rubidium dimer the discrepancy is much larger (see Table V). For the rubidium dimer, Urban and Sadlej4 _{employed an optimized}

bond length that is some 12 pm longer than the experimental bond length used in the present work, and the authors also mention that their disregard of relativistic effects may be associated with an overestimation of the bond length.4 _{As a consequence of an overestimated}

bond length, it is to be expected that the polarization along the bond axis becomes too large (i.e., an overestimated value for αzz with our choice of coordinate system). So the

fact that their CCSD(T) result4 _{for α}

zz is some 6% larger than our CCSD value for the

same tensor component is to some extent explained by the differences in bond lengths. We note, however, that there exist also a large discrepancy in the perpendicular component of the polarizability which is not explained by differences in geometry. We have determined the uncorrelated Hartree–Fock value of α(0) at the optimized bond length of 4.298 ˚A and the result is 665.1 a.u. which is some 15 a.u. larger than the value at the experimental bond length of 4.18 ˚A but still significantly lower than the Hartree–Fock value of 698.9 a.u. that is reported in Ref. 4. This clearly indicates that there is a relativistic contraction of the electron density that we account for by using relativistic ECPs in the present work. The DFT/B3PW91 method overestimates the effects of electron correlation for α(0) of the potassium and rubidium dimers but is in excellent agreement with the CCSD method when it comes to the polarizability of the two tetramers.

For the series of alkali metal dimers, a relative measure of the effects of electron correlation on the polarizability is given in Fig. 2; we compare the uncorrelated Hartree–Fock results to the corresponding CCSD results in this figure. The effects of electron correlation on the parallel components are 22%, 9%, 7%, and 1% for Li2, Na2, K2, and Rb2, respectively,

whereas, reported in the same order, the effects on the perpendicular components are −6%, −8%, −18%, and −19%. The correlation effects on the individual tensor components are thus large, and, for the alkali metal dimers, it is clear that the DFT/B3PW91 approach performs better for the perpendicular component than for the parallel component. The accuracy of the DFT results for the averaged value α(0) relies on a cancellation of errors

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between the components, and the accuracy of the DFT results for the anisotropy of the polarizability is not as high.

Since there is a square dependence between the polarizability and the dispersion coeffi-cient, one can expect the magnitude of the errors for the C6 coefficients to be twice the errors

for α(0). We estimate that the error bar of our best DFT results for the C6 coefficients of

the alkali metal clusters is 8%.

B. Dispersion interactions of alkali metal clusters and the fullerene

The dispersion coefficients for the interactions between the alkali clusters and the fullerene C60 are given in Tables II–V. The HF and DFT/B3LYP polarizabilities of C60 as given

in Ref. 30 were utilized for evaluation of the C6 coefficients in the present work. The

difference in the resulting dispersion coefficients depending on which data set is used for the polarizability of the fullerene is not significant, but we note that it was argued in Ref. 30 that the uncorrelated Hartree–Fock results for the polarizability α(iω) showed a somewhat better dispersion (i.e., frequency dependence).

In Fig. 3, we compare our best DFT results for the dispersion coefficients of the metal cluster-to-fullerene interactions against the experimental data of Kresin et al.21 _{The error}

bars of the experiment are about 30% and included in the figure. As mentioned in our previous work,22 _{the theoretical results for the smaller sodium clusters (including up to ten}

atoms) are in close agreement with experiment. In the case of larger clusters (not included in Fig. 3 but reported in the original work22_{), the agreement, although still within the}

error bars, is less convincing; there is a trend of theoretical results being smaller than the experimental counterparts. We believe that this reflects an experimental situation where there is a distribution of different metal cluster configurations with varying volume, some of which have a larger volume than the theoretically optimized volume used in the calculations. We have not pursued an investigation of the configuration dependence of the dispersion coefficients, but, if called for, it could easily be done in order to for instance correlate an accurate experimental value for the dispersion interaction with the molecular structure. We emphasize that the error bar in the theoretical results should be the same regardless of the size of the cluster, so our approach should be accurate enough to distinguish between different cluster configurations from the differences in dispersion coefficients. In addition we

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have argued in the present work that the accuracy reached in the density functional based calculations is the same for clusters involving different alkali metals, and our results should serve as appropriate reference values for future experimental work on lithium, potassium, and rubidium clusters, if such are carried out. It is apparent that the interactions between the fullerene and the clusters vary quite strongly with the species that make up the cluster; the heavier the atom the stronger the interactions, see Fig. 3.

We have also included the results for the characteristic frequency ω1[Eq. (5)] in Tables II–

V. For a given electronic structure method, a small spread of 0.01 a.u. in ω1 is found for

clusters composed of the same atom, and the mean values of ω1 are 0.090, 0.089, 0.065,

and 0.064 a.u. for the lithium, sodium, potassium, and rubidium clusters, respectively. The existence of universal characteristic frequencies for the metal clusters suggests that it is possible to construct simple structure-to-property relations for the dispersion coefficients of alkali metal clusters. We note that such a relation has also been presented for the n-alkanes in our previous work.28,32For sodium clusters, Chandrakumar et al.10showed a linear dependence between the static mean polarizability and the cluster volume. Combined with our finding of universal characteristic frequencies, one can directly find relations between the cluster volumes and the C6 coefficients with help of Eq. (5).

V. CONCLUSIONS

The complex polarization propagator approach has been shown to be an effective and direct way to determine the polarizability on the imaginary frequency axis for the metal alkali clusters. We present first-principles calculations of the electric dipole polarizabilities and the dipole-dipole dispersion coefficients of the closed-shell alkali clusters involving up to ten lithium, sodium, potassium, and rubidium atoms and the C60 fullerene. The method allows

for the employment of large and diffuse basis sets that are optimized for the calculations of the polarizabilities, and it therefore has the potential to be accurate for the determination of C6 dispersion coefficients. When benchmarked against results from wave function correlated

methods, the density functional theory results with use of the B3PW91 exchange-correlation functional are shown to be quite accurate for the isotropic polarizabilities but much less accurate for the anisotropic polarizabilities. The performance of the density functional based approach is consistent for the series of alkali metals, and the treatment of electron

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correlation in the metal clusters parallels that of the second-order Møller–Plesset method. The estimated error bars for the B3PW91 results for α(0) and the C6 coefficients are 4% and

8%, respectively. The theoretical results for dispersion interactions of the sodium clusters and the fullerene are well within the error bars of the experiment, and our set of results for the series of alkali metal cluster-to-cluster and cluster-to-fullerene dispersion interactions provides a set of theoretical reference values of consistent accuracy.

VI. ACKNOWLEDGMENT

The authors would like to express their gratitude towards Professor Vitaly V. Kresin for sharing a list of his experimental C6 dispersion coefficients, Professor Andrzej J. Sadlej

for providing a high-quality basis set for sodium, and Dr. K. R. S. Chandrakumar and his collaborators for sharing the optimized structures for lithium clusters. The authors acknowl-edge financial support from the Swedish research council (grant No. 621-2002-5546) and the European Union (Contract No. 012142-NANOCASE) as well as a grant for computing time at the National Supercomputer Centre (NSC) in Sweden.

∗ _{Electronic address: panor@ifm.liu.se}

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FIG. 1: Molecular structures for the alkali metal clusters. The bond labels refer to the optimized bond distances given in Table I.

FIG. 2: Effects of electron correlation on the static polarizabilities of the alkali metal dimers.

FIG. 3: C6 dispersion coefficients for alkali clusters containing up to 10 atoms and the fullerene

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TABLE I: B3LYP optimized geometries for the sodium, potassium and rubidium clusters. All bond lengths are given in units of ˚Angstr¨om.

n Point group Bonda _Nab

n Kn Rbn 4 D2h a 3.503 4.492 4.839 b _3.064 _4.016 _4.345 6 C5v a 3.610 4.573 4.878 b _3.437 _4.346 _4.712 8 Td a 3.476 4.486 4.810 b _3.573 _4.601 _5.062 10 D4d a 3.516 4.523 4.868 b _3.366 _4.367 _4.738 c _3.761 _4.767 _5.146

a_{The bond labels refer to the labeling made in Fig. 1.}

b_{The B3LYP/6-311+G(d) optimized structure for the tetramer was taken from Ref. 10 and the}

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TABLE II: Static mean polarizabilities (a.u.), dipole-dipole dispersion coefficients (103 _{a.u.), and}

effective frequencies (a.u.) for the Lin clusters and C60.

Lin− Lin Lin− C60

Cluster Methoda _Ref. _α

xx αyy αzz α C6 ωb1 C6c C6d Lie 2 HF 180.6 180.6 253.6 204.9 2.736 0.0869 12.01 12.23 DFT 174.1 174.1 276.5 208.2 2.747 0.0845 11.98 12.20 DFTf _169.3 _169.3 _266.4 _201.7 _2.626 _0.0861 _11.85 _12.06 MP2 15 202.8 CCSDg _169.2 _169.2 _308.7 _215.7 _2.862 _0.0820 CCSDf,g _151.6 _151.6 _312.0 _205.0 _2.661 _0.0844 CCSD(T) 4 169.2 169.2 309.7 216.0 Expt. 18 221.3 12 221.1 Li4 HF 290.8 523.1 239.5 351.1 8.198 0.0887 21.31 21.67 DFT 286.5 579.4 233.7 366.5 8.527 0.0846 21.54 21.92 DFTf _280.8 _566.1 _231.1 _359.3 _8.306 _0.0858 _21.45 _21.82 MP2 15 343.7 MP2 17 301.3 596.2 247.1 381.5 CCSD 16 284.6 485.7 228.9 333.1 CCSDg _294.0 _498.6 _234.7 _342.4 _8.062 _0.0917 CCSD(T) 17 296.4 621.4 243.2 387.0 Expt. 18 326.6 12 327.2

a_{Unless specified, Sadlej’s polarization basis set is used.}36 b_{Effective frequency is determined as ω}

1= 4C6/3 [ α(0) ] 2

.

c_{Results for C}

60are taken from Ref. 30 and obtained at the Hartree–Fock level with Sadlej’s basis set. d_{Results for C}

60 are taken from Ref. 30 and obtained at the DFT/B3LYP level with Sadlej’s basis set. e_{The experimental bond length of 2.6725 ˚}_{A is used in the present work. At the optimized bond length of}

2.709 ˚A used in the MP2 calculation, our DFT/B3PW91 result for α in the large basis set is 204.4 a.u.

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TABLE II: Continued.

Lin− Lin Lin− C60

xx αyy αzz α C6 ωb1 C6c C6d Li6 HF 603.6 313.9 609.6 509.0 17.74 0.0913 31.36 31.90 DFT 610.2 302.9 616.9 510.0 17.58 0.0901 31.31 31.85 MP2 15 507.1 Expt. 18 360.4 12 359.1 Li8 HF 608.1 608.3 608.4 608.3 25.95 0.0935 38.77 39.42 DFT 617.7 619.3 618.8 618.6 26.47 0.0922 39.06 39.71 MP2 15 609.2 CCSD 16 526.4 544.3 552.0 540.9 Expt. 18 561.5 12 559.3 Li10 HF 612.8 613.0 949.5 725.1 37.68 0.0956 47.08 47.85 DFT 605.5 605.9 995.4 735.6 38.10 0.0939 47.24 48.02 MP2 15 744.6 Expt. 18 701.8

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TABLE III: Static mean polarizabilities (a.u.), dipole-dipole dispersion coefficients (103 _{a.u.), and}

effective frequencies (a.u.) for the Nan clusters and C60.

Nan− Nan Nan− C60

xx αyy αzz α C6 ωb1 C6c C6d Na2 HF 229.9 229.9 358.2 272.6 4.681 0.0840 15.77 16.02 DFT 204.9 204.9 352.3 254.0 4.187 0.0865 15.10 15.36 DFTe _204.4 _204.4 _350.1 _252.9 _4.174 _0.0870 _15.14 _15.41 MP2 11 252.5 CCSDf _210.6 _210.6 _391.4 _270.9 _4.659 _0.0846 CCSDe,f _189.1 _189.1 _400.4 _259.5 _4.362 _0.0864 CCSD(T) 11 263.3 Expt. 3 255.8 12 265.2 21 17.62 Na4 HF 416.9 839.2 334.2 530.1 17.06 0.0810 30.19 30.72 DFT 388.8 833.8 304.9 509.2 15.98 0.0822 29.43 29.94 DFTe _385.4 _830.9 _308.1 _508.2 _15.99 _0.0826 _29.55 _30.06 MP2 10 508.6 CCSDf _407.5 _809.5 _317.5 _511.5 _16.80 _0.0856 CCSD(T) 10 509.6 Expt. 3 545.9 12 565.5 21 26.56

1= 4C6/3 [ α(0) ]2. c_{Results for C}

60 are taken from Ref. 30 and obtained at the DFT/B3LYP level with Sadlej’s basis set. e_{Calculated with a large [19s15p12d6f ] basis set.}37

f_C

6 result is obtained from the mean value of the lower [n,n-1]α and upper [n,n-1]β Pad´e

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TABLE III: Continued.

Nan− Nan Nan− C60

xx αyy αzz α C6 ωb1 C6c C6d Na6 HF 885.8 457.2 888.8 743.9 35.55 0.0856 44.09 44.85 DFT 838.9 418.2 841.9 699.7 32.47 0.0884 42.60 43.33 Expt. 3 823.7 12 754.3 21 38.91 Na8 HF 883.9 883.9 883.9 883.9 52.68 0.0899 54.71 55.63 DFT 845.9 845.9 845.9 845.9 49.47 0.0922 53.44 54.33 Expt. 3 880.4 12 901.0 14 955.6 21 55.01 Na10 HF 867.3 867.3 1425 1053 76.60 0.0921 66.53 67.63 DFT 800.7 800.7 1397 999.4 70.88 0.0946 64.62 65.67 Expt. 3 1296 21 63.71

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TABLE IV: Static mean polarizabilities (a.u.), dipole-dipole dispersion coefficients (103 _{a.u.), and}

effective frequencies (a.u.) for the Kn clusters and C60.

Kn− Kn Kn− C60

xx αyy αzz α C6 ω1b C6c C6d K2 HF 465.4 465.4 736.7 555.8 14.17 0.0612 25.79 26.28 DFT 389.9 389.9 694.5 491.4 11.81 0.0652 24.20 24.64 DFTe _396.1 _396.1 _700.2 _497.5 _12.00 _0.0646 _24.32 _24.76 CCSDf _380.5 _380.5 _789.6 _516.9 _13.15 _0.0656 CCSD(T) 4 381.8 381.8 758.8 507.5 Expt. 3 485.9 19 519.6±40.5 K4 HF 858.6 1705 688.8 1084 52.02 0.0590 49.48 50.41 DFT 776.1 1659 599.5 1012 46.77 0.0610 47.69 48.57 DFTe _769.3 ₁₆₆₂ _606.5 ₁₀₁₃ _46.77 _0.0608 _47.68 _48.55 CCSDf _806.8 ₁₆₂₉ _608.0 ₁₀₁₅ _50.20 _0.0650 K6 HF 1788 914.6 1788 1497 105.4 0.0627 71.75 73.07 DFT 1636 805.5 1637 1359 91.91 0.0663 68.49 69.72 K8 HF 1838 1838 1838 1838 164.0 0.0647 90.96 92.59 DFT 1710 1710 1710 1710 148.3 0.0676 87.96 89.51 K10 HF 1791 1791 2851 2145 232.7 0.0674 109.9 111.9 DFT 1598 1598 2751 1983 208.0 0.0706 105.9 107.7

1= 4C6/3 [ α(0) ]2. c_{Results for C}

60 are taken from Ref. 30 and obtained at the DFT/B3LYP level with Sadlej’s basis set. e_{Calculated with a large [19s15p12d6f] basis set.}

f_C

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TABLE V: Static mean polarizabilities (a.u.), dipole-dipole dispersion coefficients (103 _{a.u.), and}

effective frequencies (a.u.) for the Rbn clusters and C60.

Rbn− Rbn Rbn− C60

xx αyy αzz α C6 ω1b C6c C6d Rb2 HF 533.0 533.0 883.8 649.9 18.74 0.0591 29.87 30.43 DFT 434.6 434.6 793.8 554.3 14.93 0.0648 27.66 28.15 DFTe _447.7 _447.7 _801.5 _565.6 _15.32 _0.0639 _27.87 _28.37 CCSDf _429.4 _429.4 _888.5 _582.4 _16.70 _0.0656 CCSD(T) 4 471.2 471.2 941.7 628.0 Expt. 19 533.1±40.5 Rb4 HF 1033 2136 820.0 1330 73.64 0.0555 58.65 59.74 DFT 909.8 2007 699.7 1206 63.85 0.0586 55.96 56.97 DFTe _907.5 ₁₉₉₈ _711.0 ₁₂₀₅ _63.80 _0.0585 _55.90 _56.91 CCSDf _958.0 ₂₀₃₆ _716.2 ₁₂₃₇ _70.78 _0.0617 Rb6 HF 2136 1099 2136 1790 144.3 0.0600 84.22 85.74 DFT 1912 945.5 1912 1590 122.3 0.0645 79.79 81.18 Rb8 HF 2227 2227 2227 2227 228.8 0.0615 107.5 109.5 DFT 2009 2009 2009 2009 198.8 0.0656 102.7 104.5 Rb10 HF 2125 2125 3537 2596 326.5 0.0646 130.7 133.0 DFT 1891 1891 3334 2372 287.6 0.0682 125.4 127.5